In mathematics, a **Seifert surface** (named after German mathematician Herbert Seifert) is a surface whose boundary is a given knot or link.

Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface. Seifert surfaces are also interesting in their own right, and the subject of considerable research.

Specifically, let *L* be a tame oriented knot or link in Euclidean 3-space (or in the 3-sphere). A **Seifert surface** is a compact, connected, oriented surface *S* embedded in 3-space whose boundary is *L* such that the orientation on *L* is just the induced orientation from *S*, and every connected component of *S* has non-empty boundary.

Note that any compact, connected, oriented surface with nonempty boundary in Euclidean 3-space is the Seifert surface associated to its boundary link. A single knot or link can have many different inequivalent Seifert surfaces. A Seifert surface must be oriented. It is possible to associate unoriented (and not necessarily orientable) surfaces to knots as well.

Read more about Seifert Surface: Examples, Existence and Seifert Matrix, Genus of A Knot

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### Famous quotes containing the word surface:

“It was a pretty game, played on the smooth *surface* of the pond, a man against a loon.”

—Henry David Thoreau (1817–1862)